Use transformations of the graph of $$ y=1 / x $$ to graph the rational function, as in Erample 2 . $$ s(x)=\frac{3}{x+1} $$

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To graph the rational function $$ s(x) = \frac{3}{x+1} $$ using transformations of the graph of $$ y = \frac{1}{x} $$, we can follow these steps:

1. **Identify the parent function**: The parent function is $$ y = \frac{1}{x} $$.

2. **Horizontal Shift**: The function $$ s(x) $$ has $$ x + 1 $$ in the denominator. This indicates a horizontal shift to the left by 1 unit. Therefore, the graph of $$ y = \frac{1}{x} $$ will be shifted left to become $$ y = \frac{1}{x + 1} $$.

3. **Vertical Stretch**: The coefficient 3 in the numerator indicates a vertical stretch by a factor of 3. Thus, the graph of $$ y = \frac{1}{x + 1} $$ will be stretched vertically to become $$ y = \frac{3}{x + 1} $$.

4. **Asymptotes**:
   - **Vertical Asymptote**: The vertical asymptote occurs where the denominator is zero. For $$ s(x) = \frac{3}{x + 1} $$, set $$ x + 1 = 0 $$ which gives $$ x = -1 $$.
   - **Horizontal Asymptote**: As $$ x $$ approaches infinity or negative infinity, $$ s(x) $$ approaches 0. Therefore, the horizontal asymptote is $$ y = 0 $$.

5. **Plotting Key Points**: 
   - When $$ x = -2 $$, $$ s(-2) = \frac{3}{-2 + 1} = \frac{3}{-1} = -3 $$.
   - When $$ x = 0 $$, $$ s(0) = \frac{3}{0 + 1} = 3 $$.
   - When $$ x = 1 $$, $$ s(1) = \frac{3}{1 + 1} = \frac{3}{2} = 1.5 $$.

6. **Sketch the Graph**: 
   - Start by drawing the vertical asymptote at $$ x = -1 $$ and the horizontal asymptote at $$ y = 0 $$.
   - Plot the points calculated: $$ (-2, -3) $$, $$ (0, 3) $$, and $$ (1, 1.5) $$.
   - Draw the curve approaching the asymptotes, ensuring it reflects the behavior of the parent function.

By following these steps, you can accurately graph the function $$ s(x) = \frac{3}{x + 1} $$ based on transformations of the graph of $$ y = \frac{1}{x} $$.
To graph $$ s(x) = \frac{3}{x+1} $$, start with the parent function $$ y = \frac{1}{x} $$. Shift it left by 1 unit and stretch it vertically by 3. Draw the vertical asymptote at $$ x = -1 $$ and horizontal asymptote at $$ y = 0 $$. Plot key points like $$ (-2, -3) $$, $$ (0, 3) $$, and $$ (1, 1.5) $$, and sketch the curve accordingly.

Use transformations of the graph of \( y=1 / x \) to graph the rational function, as in Erample 2 . \[ s(x)=\frac{3}{x+1} \]

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